F. Amato

Finite-time control of discrete-time linear systems

In this note, we consider the finite-time stabilization of discrete-time linear systems subject to disturbances generated by an exosystem. Finite-time stability can be used in all those applications where large values of the state should not be attained, for instance in the presence of saturations. The main result provided in the note is a sufficient condition for finite-time stabilization via state feedback. This result is then used to find some sufficient conditions for the existence of an output feedback controller guaranteeing finite-time stability. All the conditions are then reduced to feasibility problems involving linear matrix inequalities (LMIs). Some numerical examples are presented to illustrate the proposed methodology.

New sufficient conditions for the stability of slowly varying linear systems

The frozen-time approach is used to state some new sufficient conditions for the stability of linear time-varying systems. An upper bound on the norm of the time derivative of system matrix which, under different assumptions on frozen-time system eigenvalues, guarantees asymptotic stability or exponential stability of the system is established. >

Stabilization of Bilinear Systems Via Linear State-Feedback Control

This paper deals with the problem of stabilizing a bilinear system via linear state-feedback control. The proposed procedures enable us to compute a static state-feedback controller such that the zero-equilibrium point of the closed-loop system is asymptotically stable; moreover, it ensure that an assigned polytopic region is enclosed into the domain of attraction of the equilibrium point. The controller design requires the solution of a convex optimization problem involving linear matrix inequalities. The applicability of the technique is illustrated through an example, dealing with the design of a controller for a Cuk dc-dc converter.

Necessary and sufficient conditions for finite-time stability of linear systems

In this paper we consider the finite-time stability and finite-time boundedness problems for linear systems subject to exogenous disturbances. The main results of the paper are some necessary and sufficient conditions, obtained by means of an approach based on operator theory; such conditions improve some recent results on this topic. An example is provided to illustrate the proposed technique.

An insight into tumor dormancy equilibrium via the analysis of its domain of attraction

Abstract The trajectories of the dynamic system which regulates the competition between the populations of malignant cells and immune cells may tend to an asymptotically stable equilibrium in which the sizes of these populations do not vary, which is called tumor dormancy. Especially for lower steady-state sizes of the population of malignant cells, this equilibrium represents a desirable clinical condition since the tumor growth is blocked. In this context, it is of mandatory importance to analyze the robustness of this clinical favorable state of health in the face of perturbations. To this end, the paper presents an optimization technique to determine whether an assigned rectangular region, which surrounds an asymptotically stable equilibrium point of a quadratic systems, is included into the domain of attraction of the equilibrium itself. The biological relevance of the application of this technique to the analysis of tumor growth dynamics is shown on the basis of a recent quadratic model of the tumor–immune system competition dynamics. Indeed the application of the proposed methodology allows to ensure that a given safety region, determined on the basis of clinical considerations, belongs to the domain of attraction of the tumor blocked equilibrium; therefore for the set of perturbed initial conditions which belong to such region, the convergence to the healthy steady state is guaranteed. The proposed methodology can also provide an optimal strategy for cancer treatment.

Finite-time control of linear time-varying systems via output feedback

This paper deals with various finite-time analysis and design problems for continuous-time time-varying linear systems. We present some necessary and sufficient conditions for finite-time stability and then we turn to the design problem. In this context, we consider both the state feedback and the output feedback problems. For both cases, we end up with some sufficient conditions involving linear differential matrix inequalities.

A note on quadratic stability of uncertain linear discrete-time systems

In this paper we consider a linear discrete-time system depending on a vector of uncertain parameters. Assuming that the system matrix depends on parameters as the ratio of a multiaffine matrix-valued function and a multiaffine polynomial and that the parameters range in a hyper-rectangle, we show that the uncertain system is quadratically stable if and and only if the set of vertex systems is quadratically stable. This allows us to state a necessary and sufficient condition for quadratic stability in terms of the solvability of a feasibility problem involving linear matrix inequalities.

Output feedback control of nonlinear quadratic systems

This paper provides some sufficient conditions for the stabilization of nonlinear quadratic systems via output feedback. The main contribution consists of a design procedure which enables to find a dynamic output feedback controller guaranteeing for the closed-loop system: i) the local asymptotic stability of the zero equilibrium point; ii) the inclusion of a given polytopic region into the domain of attraction of the zero equilibrium point. This design procedure is formulated in terms of a Linear Matrix Inequalities (LMIs) feasibility problem, which can be efficiently solved via available optimization algorithms. The effectiveness of the proposed methodology is shown through a numerical example.

Stabilization of bilinear systems via linear state feedback control

In this paper we consider the problem of stabilizing a bilinear system via linear state feedback control. A procedure is proposed which, given a polytope V surrounding the origin of the state space, finds, if existing, a controller in the form u = Kx, such that the zero equilibrium point of the closed loop system is asymptotically stable and V is enclosed into the domain of attraction of the equilibrium. The controller design requires the solution of a convex optimization problem involving linear matrix inequalities. An example illustrates the applicability of the proposed technique.

Finite-time control for uncertain linear systems with disturbance inputs

We consider the static output feedback, finite-time disturbance rejection problem for linear systems with time-varying norm-bounded uncertainties. The first result provided in the paper is a sufficient condition for finite-time state feedback disturbance rejection in the presence of constant disturbances. This condition requires the solution of an LMI. Then we consider the more general output feedback case, which is shown to be reducible to the solution of an optimization problem involving bilinear matrix inequalities. Finally we deal with the case in which the disturbance is time-varying and generated by a linear system.